Relying on weak cross-Kerr nonlinearities, we propose a nearly deterministic generation scheme of the three-photon polarization-entangled perfect $W$ state which can be applied to the perfect teleportation of an unknown single-photon state and has robust entanglement against the loss of one photon of them. Three photons entangle together by virtue of the bus function of the coherent state serving as the intermediate among them. In the scheme, three processes are executed successively and two kinds of modules are inserted into the circuit, where the homodyne measurement and the photon number measurement are aptly performed. By means of classical feedforward techniques, single-photon unitary transformation operations are performed on the corresponding photons based on the obtained measurement outcomes, by which the generation efficiency of the perfect $W$ state aims to nearly unity. Moreover, some currently available optical elements are applied in the generation process, which offer facilities for the practical implementation.

With the assistance of weak cross-Kerr nonlinearities, we present a preparation scheme of four-photon polarization-entangled decoherence-free states, which can be used to construct the minimal optical decoherence-free subspaces where a logical qubit is fully protected against collective decoherence. To complete the preparation task, one spatial entanglement process, two polarization entanglement processes, and one detecting process are applied. The fulfillments of the above processes are contributed by a cross-Kerr nonlinear interaction between the signal photons and a coherent state via Kerr media. Exploiting the available single-photon resource and simple linear optics elements, this scheme is feasible and desirable to be extended to the construction of multiphoton decoherence-free states against the collective decoherence.

A Toffoli gate plays a critical role in many quantum algorithms due to its function as a building block, which is a fundamental element for feasible large-scale quantum computation. With the help of polarization, spatial, and temporal degrees of freedom (DOFs), a construction scheme of a nearly deterministic polarization Toffoli gate is proposed, where only two two-photon gates are required. The simple construction circuit together with available techniques and optical elements facilitate the realization of the scheme presented here. This construction scheme can be utilized as a reference for multiqubit quantum gates with multiple DOFs.

We propose a construction scheme of a two-photon polarization controlled arbitrary phase gate based on weak cross-phase modulation. Assisted with weak cross-phase modulation and homodyne measurement on the coherent states, the individual photons are entangled together. Employing the combination of optical elements and classical feed-forward techniques, the target photon can have a conditionally shifted arbitrary phase with efficiency approaching nearly unity. With a large-amplitude coherent state, the high success probability of the controlled arbitrary phase gate can be guaranteed.